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Theorem bcthlem2 24842
Description: Lemma for bcth 24846. The balls in the sequence form an inclusion chain. (Contributed by Mario Carneiro, 7-Jan-2014.)
Hypotheses
Ref Expression
bcth.2 𝐽 = (MetOpenβ€˜π·)
bcthlem.4 (πœ‘ β†’ 𝐷 ∈ (CMetβ€˜π‘‹))
bcthlem.5 𝐹 = (π‘˜ ∈ β„•, 𝑧 ∈ (𝑋 Γ— ℝ+) ↦ {⟨π‘₯, π‘ŸβŸ© ∣ ((π‘₯ ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ+) ∧ (π‘Ÿ < (1 / π‘˜) ∧ ((clsβ€˜π½)β€˜(π‘₯(ballβ€˜π·)π‘Ÿ)) βŠ† (((ballβ€˜π·)β€˜π‘§) βˆ– (π‘€β€˜π‘˜))))})
bcthlem.6 (πœ‘ β†’ 𝑀:β„•βŸΆ(Clsdβ€˜π½))
bcthlem.7 (πœ‘ β†’ 𝑅 ∈ ℝ+)
bcthlem.8 (πœ‘ β†’ 𝐢 ∈ 𝑋)
bcthlem.9 (πœ‘ β†’ 𝑔:β„•βŸΆ(𝑋 Γ— ℝ+))
bcthlem.10 (πœ‘ β†’ (π‘”β€˜1) = ⟨𝐢, π‘…βŸ©)
bcthlem.11 (πœ‘ β†’ βˆ€π‘˜ ∈ β„• (π‘”β€˜(π‘˜ + 1)) ∈ (π‘˜πΉ(π‘”β€˜π‘˜)))
Assertion
Ref Expression
bcthlem2 (πœ‘ β†’ βˆ€π‘› ∈ β„• ((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1))) βŠ† ((ballβ€˜π·)β€˜(π‘”β€˜π‘›)))
Distinct variable groups:   π‘˜,𝑛,π‘Ÿ,π‘₯,𝑧   𝐢,π‘Ÿ,π‘₯   𝑔,π‘˜,𝑛,π‘Ÿ,π‘₯,𝑧,𝐷   𝑔,𝐹,π‘˜,𝑛,π‘Ÿ,π‘₯,𝑧   𝑔,𝐽,π‘˜,𝑛,π‘Ÿ,π‘₯,𝑧   𝑔,𝑀,π‘˜,𝑛,π‘Ÿ,π‘₯,𝑧   πœ‘,π‘˜,𝑛,π‘Ÿ,π‘₯,𝑧   π‘₯,𝑅   𝑔,𝑋,π‘˜,𝑛,π‘Ÿ,π‘₯,𝑧
Allowed substitution hints:   πœ‘(𝑔)   𝐢(𝑧,𝑔,π‘˜,𝑛)   𝑅(𝑧,𝑔,π‘˜,𝑛,π‘Ÿ)
Proof of Theorem bcthlem2
StepHypRef Expression
1 bcthlem.11 . . . . 5 (πœ‘ β†’ βˆ€π‘˜ ∈ β„• (π‘”β€˜(π‘˜ + 1)) ∈ (π‘˜πΉ(π‘”β€˜π‘˜)))
2 fvoveq1 7432 . . . . . . 7 (π‘˜ = 𝑛 β†’ (π‘”β€˜(π‘˜ + 1)) = (π‘”β€˜(𝑛 + 1)))
3 id 22 . . . . . . . 8 (π‘˜ = 𝑛 β†’ π‘˜ = 𝑛)
4 fveq2 6892 . . . . . . . 8 (π‘˜ = 𝑛 β†’ (π‘”β€˜π‘˜) = (π‘”β€˜π‘›))
53, 4oveq12d 7427 . . . . . . 7 (π‘˜ = 𝑛 β†’ (π‘˜πΉ(π‘”β€˜π‘˜)) = (𝑛𝐹(π‘”β€˜π‘›)))
62, 5eleq12d 2828 . . . . . 6 (π‘˜ = 𝑛 β†’ ((π‘”β€˜(π‘˜ + 1)) ∈ (π‘˜πΉ(π‘”β€˜π‘˜)) ↔ (π‘”β€˜(𝑛 + 1)) ∈ (𝑛𝐹(π‘”β€˜π‘›))))
76rspccva 3612 . . . . 5 ((βˆ€π‘˜ ∈ β„• (π‘”β€˜(π‘˜ + 1)) ∈ (π‘˜πΉ(π‘”β€˜π‘˜)) ∧ 𝑛 ∈ β„•) β†’ (π‘”β€˜(𝑛 + 1)) ∈ (𝑛𝐹(π‘”β€˜π‘›)))
81, 7sylan 581 . . . 4 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (π‘”β€˜(𝑛 + 1)) ∈ (𝑛𝐹(π‘”β€˜π‘›)))
9 bcthlem.9 . . . . . 6 (πœ‘ β†’ 𝑔:β„•βŸΆ(𝑋 Γ— ℝ+))
109ffvelcdmda 7087 . . . . 5 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (π‘”β€˜π‘›) ∈ (𝑋 Γ— ℝ+))
11 bcth.2 . . . . . . 7 𝐽 = (MetOpenβ€˜π·)
12 bcthlem.4 . . . . . . 7 (πœ‘ β†’ 𝐷 ∈ (CMetβ€˜π‘‹))
13 bcthlem.5 . . . . . . 7 𝐹 = (π‘˜ ∈ β„•, 𝑧 ∈ (𝑋 Γ— ℝ+) ↦ {⟨π‘₯, π‘ŸβŸ© ∣ ((π‘₯ ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ+) ∧ (π‘Ÿ < (1 / π‘˜) ∧ ((clsβ€˜π½)β€˜(π‘₯(ballβ€˜π·)π‘Ÿ)) βŠ† (((ballβ€˜π·)β€˜π‘§) βˆ– (π‘€β€˜π‘˜))))})
1411, 12, 13bcthlem1 24841 . . . . . 6 ((πœ‘ ∧ (𝑛 ∈ β„• ∧ (π‘”β€˜π‘›) ∈ (𝑋 Γ— ℝ+))) β†’ ((π‘”β€˜(𝑛 + 1)) ∈ (𝑛𝐹(π‘”β€˜π‘›)) ↔ ((π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+) ∧ (2nd β€˜(π‘”β€˜(𝑛 + 1))) < (1 / 𝑛) ∧ ((clsβ€˜π½)β€˜((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1)))) βŠ† (((ballβ€˜π·)β€˜(π‘”β€˜π‘›)) βˆ– (π‘€β€˜π‘›)))))
1514expr 458 . . . . 5 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ ((π‘”β€˜π‘›) ∈ (𝑋 Γ— ℝ+) β†’ ((π‘”β€˜(𝑛 + 1)) ∈ (𝑛𝐹(π‘”β€˜π‘›)) ↔ ((π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+) ∧ (2nd β€˜(π‘”β€˜(𝑛 + 1))) < (1 / 𝑛) ∧ ((clsβ€˜π½)β€˜((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1)))) βŠ† (((ballβ€˜π·)β€˜(π‘”β€˜π‘›)) βˆ– (π‘€β€˜π‘›))))))
1610, 15mpd 15 . . . 4 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ ((π‘”β€˜(𝑛 + 1)) ∈ (𝑛𝐹(π‘”β€˜π‘›)) ↔ ((π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+) ∧ (2nd β€˜(π‘”β€˜(𝑛 + 1))) < (1 / 𝑛) ∧ ((clsβ€˜π½)β€˜((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1)))) βŠ† (((ballβ€˜π·)β€˜(π‘”β€˜π‘›)) βˆ– (π‘€β€˜π‘›)))))
178, 16mpbid 231 . . 3 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ ((π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+) ∧ (2nd β€˜(π‘”β€˜(𝑛 + 1))) < (1 / 𝑛) ∧ ((clsβ€˜π½)β€˜((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1)))) βŠ† (((ballβ€˜π·)β€˜(π‘”β€˜π‘›)) βˆ– (π‘€β€˜π‘›))))
18 cmetmet 24803 . . . . . . . . . . . 12 (𝐷 ∈ (CMetβ€˜π‘‹) β†’ 𝐷 ∈ (Metβ€˜π‘‹))
1912, 18syl 17 . . . . . . . . . . 11 (πœ‘ β†’ 𝐷 ∈ (Metβ€˜π‘‹))
20 metxmet 23840 . . . . . . . . . . 11 (𝐷 ∈ (Metβ€˜π‘‹) β†’ 𝐷 ∈ (∞Metβ€˜π‘‹))
2119, 20syl 17 . . . . . . . . . 10 (πœ‘ β†’ 𝐷 ∈ (∞Metβ€˜π‘‹))
2211mopntop 23946 . . . . . . . . . 10 (𝐷 ∈ (∞Metβ€˜π‘‹) β†’ 𝐽 ∈ Top)
2321, 22syl 17 . . . . . . . . 9 (πœ‘ β†’ 𝐽 ∈ Top)
24 xp1st 8007 . . . . . . . . . . . 12 ((π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+) β†’ (1st β€˜(π‘”β€˜(𝑛 + 1))) ∈ 𝑋)
25 xp2nd 8008 . . . . . . . . . . . . 13 ((π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+) β†’ (2nd β€˜(π‘”β€˜(𝑛 + 1))) ∈ ℝ+)
2625rpxrd 13017 . . . . . . . . . . . 12 ((π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+) β†’ (2nd β€˜(π‘”β€˜(𝑛 + 1))) ∈ ℝ*)
2724, 26jca 513 . . . . . . . . . . 11 ((π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+) β†’ ((1st β€˜(π‘”β€˜(𝑛 + 1))) ∈ 𝑋 ∧ (2nd β€˜(π‘”β€˜(𝑛 + 1))) ∈ ℝ*))
28 blssm 23924 . . . . . . . . . . . 12 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ (1st β€˜(π‘”β€˜(𝑛 + 1))) ∈ 𝑋 ∧ (2nd β€˜(π‘”β€˜(𝑛 + 1))) ∈ ℝ*) β†’ ((1st β€˜(π‘”β€˜(𝑛 + 1)))(ballβ€˜π·)(2nd β€˜(π‘”β€˜(𝑛 + 1)))) βŠ† 𝑋)
29283expb 1121 . . . . . . . . . . 11 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ ((1st β€˜(π‘”β€˜(𝑛 + 1))) ∈ 𝑋 ∧ (2nd β€˜(π‘”β€˜(𝑛 + 1))) ∈ ℝ*)) β†’ ((1st β€˜(π‘”β€˜(𝑛 + 1)))(ballβ€˜π·)(2nd β€˜(π‘”β€˜(𝑛 + 1)))) βŠ† 𝑋)
3021, 27, 29syl2an 597 . . . . . . . . . 10 ((πœ‘ ∧ (π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+)) β†’ ((1st β€˜(π‘”β€˜(𝑛 + 1)))(ballβ€˜π·)(2nd β€˜(π‘”β€˜(𝑛 + 1)))) βŠ† 𝑋)
31 df-ov 7412 . . . . . . . . . . . 12 ((1st β€˜(π‘”β€˜(𝑛 + 1)))(ballβ€˜π·)(2nd β€˜(π‘”β€˜(𝑛 + 1)))) = ((ballβ€˜π·)β€˜βŸ¨(1st β€˜(π‘”β€˜(𝑛 + 1))), (2nd β€˜(π‘”β€˜(𝑛 + 1)))⟩)
32 1st2nd2 8014 . . . . . . . . . . . . 13 ((π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+) β†’ (π‘”β€˜(𝑛 + 1)) = ⟨(1st β€˜(π‘”β€˜(𝑛 + 1))), (2nd β€˜(π‘”β€˜(𝑛 + 1)))⟩)
3332fveq2d 6896 . . . . . . . . . . . 12 ((π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+) β†’ ((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1))) = ((ballβ€˜π·)β€˜βŸ¨(1st β€˜(π‘”β€˜(𝑛 + 1))), (2nd β€˜(π‘”β€˜(𝑛 + 1)))⟩))
3431, 33eqtr4id 2792 . . . . . . . . . . 11 ((π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+) β†’ ((1st β€˜(π‘”β€˜(𝑛 + 1)))(ballβ€˜π·)(2nd β€˜(π‘”β€˜(𝑛 + 1)))) = ((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1))))
3534adantl 483 . . . . . . . . . 10 ((πœ‘ ∧ (π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+)) β†’ ((1st β€˜(π‘”β€˜(𝑛 + 1)))(ballβ€˜π·)(2nd β€˜(π‘”β€˜(𝑛 + 1)))) = ((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1))))
3611mopnuni 23947 . . . . . . . . . . . 12 (𝐷 ∈ (∞Metβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
3721, 36syl 17 . . . . . . . . . . 11 (πœ‘ β†’ 𝑋 = βˆͺ 𝐽)
3837adantr 482 . . . . . . . . . 10 ((πœ‘ ∧ (π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+)) β†’ 𝑋 = βˆͺ 𝐽)
3930, 35, 383sstr3d 4029 . . . . . . . . 9 ((πœ‘ ∧ (π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+)) β†’ ((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1))) βŠ† βˆͺ 𝐽)
40 eqid 2733 . . . . . . . . . 10 βˆͺ 𝐽 = βˆͺ 𝐽
4140sscls 22560 . . . . . . . . 9 ((𝐽 ∈ Top ∧ ((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1))) βŠ† βˆͺ 𝐽) β†’ ((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1))) βŠ† ((clsβ€˜π½)β€˜((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1)))))
4223, 39, 41syl2an2r 684 . . . . . . . 8 ((πœ‘ ∧ (π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+)) β†’ ((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1))) βŠ† ((clsβ€˜π½)β€˜((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1)))))
43 difss2 4134 . . . . . . . 8 (((clsβ€˜π½)β€˜((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1)))) βŠ† (((ballβ€˜π·)β€˜(π‘”β€˜π‘›)) βˆ– (π‘€β€˜π‘›)) β†’ ((clsβ€˜π½)β€˜((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1)))) βŠ† ((ballβ€˜π·)β€˜(π‘”β€˜π‘›)))
44 sstr2 3990 . . . . . . . 8 (((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1))) βŠ† ((clsβ€˜π½)β€˜((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1)))) β†’ (((clsβ€˜π½)β€˜((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1)))) βŠ† ((ballβ€˜π·)β€˜(π‘”β€˜π‘›)) β†’ ((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1))) βŠ† ((ballβ€˜π·)β€˜(π‘”β€˜π‘›))))
4542, 43, 44syl2im 40 . . . . . . 7 ((πœ‘ ∧ (π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+)) β†’ (((clsβ€˜π½)β€˜((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1)))) βŠ† (((ballβ€˜π·)β€˜(π‘”β€˜π‘›)) βˆ– (π‘€β€˜π‘›)) β†’ ((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1))) βŠ† ((ballβ€˜π·)β€˜(π‘”β€˜π‘›))))
4645a1d 25 . . . . . 6 ((πœ‘ ∧ (π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+)) β†’ ((2nd β€˜(π‘”β€˜(𝑛 + 1))) < (1 / 𝑛) β†’ (((clsβ€˜π½)β€˜((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1)))) βŠ† (((ballβ€˜π·)β€˜(π‘”β€˜π‘›)) βˆ– (π‘€β€˜π‘›)) β†’ ((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1))) βŠ† ((ballβ€˜π·)β€˜(π‘”β€˜π‘›)))))
4746ex 414 . . . . 5 (πœ‘ β†’ ((π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+) β†’ ((2nd β€˜(π‘”β€˜(𝑛 + 1))) < (1 / 𝑛) β†’ (((clsβ€˜π½)β€˜((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1)))) βŠ† (((ballβ€˜π·)β€˜(π‘”β€˜π‘›)) βˆ– (π‘€β€˜π‘›)) β†’ ((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1))) βŠ† ((ballβ€˜π·)β€˜(π‘”β€˜π‘›))))))
48473impd 1349 . . . 4 (πœ‘ β†’ (((π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+) ∧ (2nd β€˜(π‘”β€˜(𝑛 + 1))) < (1 / 𝑛) ∧ ((clsβ€˜π½)β€˜((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1)))) βŠ† (((ballβ€˜π·)β€˜(π‘”β€˜π‘›)) βˆ– (π‘€β€˜π‘›))) β†’ ((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1))) βŠ† ((ballβ€˜π·)β€˜(π‘”β€˜π‘›))))
4948adantr 482 . . 3 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (((π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+) ∧ (2nd β€˜(π‘”β€˜(𝑛 + 1))) < (1 / 𝑛) ∧ ((clsβ€˜π½)β€˜((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1)))) βŠ† (((ballβ€˜π·)β€˜(π‘”β€˜π‘›)) βˆ– (π‘€β€˜π‘›))) β†’ ((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1))) βŠ† ((ballβ€˜π·)β€˜(π‘”β€˜π‘›))))
5017, 49mpd 15 . 2 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ ((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1))) βŠ† ((ballβ€˜π·)β€˜(π‘”β€˜π‘›)))
5150ralrimiva 3147 1 (πœ‘ β†’ βˆ€π‘› ∈ β„• ((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1))) βŠ† ((ballβ€˜π·)β€˜(π‘”β€˜π‘›)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062   βˆ– cdif 3946   βŠ† wss 3949  βŸ¨cop 4635  βˆͺ cuni 4909   class class class wbr 5149  {copab 5211   Γ— cxp 5675  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409   ∈ cmpo 7411  1st c1st 7973  2nd c2nd 7974  1c1 11111   + caddc 11113  β„*cxr 11247   < clt 11248   / cdiv 11871  β„•cn 12212  β„+crp 12974  βˆžMetcxmet 20929  Metcmet 20930  ballcbl 20931  MetOpencmopn 20934  Topctop 22395  Clsdccld 22520  clsccl 22522  CMetccmet 24771
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187  ax-pre-sup 11188
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-er 8703  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-sup 9437  df-inf 9438  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-div 11872  df-nn 12213  df-2 12275  df-n0 12473  df-z 12559  df-uz 12823  df-q 12933  df-rp 12975  df-xneg 13092  df-xadd 13093  df-xmul 13094  df-topgen 17389  df-psmet 20936  df-xmet 20937  df-met 20938  df-bl 20939  df-mopn 20940  df-top 22396  df-topon 22413  df-bases 22449  df-cld 22523  df-cls 22525  df-cmet 24774
This theorem is referenced by:  bcthlem3  24843  bcthlem4  24844
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